Foliations : Whats next after Thurston ? The mathematical legacy of - - PowerPoint PPT Presentation

Foliations : What’s next after Thurston ?

The mathematical legacy of Bill Thurston, Étienne Ghys, CNRS ENS Lyon

A dozen publications between 1972 and 1976

On proofs and progress in mathematics, Thurston 1994

“First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate

• student. [...]

I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a

• foliation. I proved a number of other significant theorems.

I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog.”

Foliations ?

“An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take

• place. I heard from a number of mathematicians that they

were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it

• ut. People told me (not as a complaint, but as a

compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.”

Foliations

Codimension q foliation on a manifold X : An open covering Ui of X. Submersions fi : Ui → Rq. A cocycle θi,j of C ∞ diffeomorphisms between open sets of Rq such that θj,k ◦ θi,j = θi,k where it is defined and fj = θi,j ◦ fi.

Leçons de Stockholm (1895)

The Reeb component (1948)

1895 : Leçons de Stockholm (Painlevé). 1944-1948 : Foliation on the 3-sphere (Reeb). 1955-1958 : Inexistence of codimension 1 analytic foliations on spheres (Haefliger). 1964 : Every codimension 1 foliation on the 3-sphere has a compact leaf (Novikov). 1968 : Topological obstruction to integrability : certain plane fields are not homotopic to a foliation (Bott). 1970 : Classifying space BΓ (Haefliger).

Thurston’s helical wobble (1971)

Helical wobble

On proofs and progress in mathematics, Thurston 1994

“I threw out prize cryptic tidbits of insight, such as “the Godbillon-Vey invariant measures the helical wobble of a foliation”, that remained mysterious to most mathematicans who read them. This created a high entry barrier : I think many graduate students and mathematicians were discouraged that it was hard to learn and understand the proofs of key theorems.”

Helical wobble

Alejandra Ruddoff “Diacronia” 2005

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Godbillon-Vey invariant (1971)

A (transversaly orientable) codimension 1 foliation F on M is defined by a 1-form ω. Integrability of F implies ω ∧ dω = 0. There exists α such that dω = ω ∧ α. The 3-form α ∧ dα is closed. Its cohomology class in H3(M, R) is independent of all choices : this is the Godbillon-Vey invariant of F. If dim(M) = 3 and if M is oriented, this is a number : gv(F) ∈ R. Two cobordant foliations have the same Godbillon-Vey number.

Theorem (Thurston 1971) :

There exists of family Fλ of foliations on S3 such that gv(Fλ) varies continuously.

Helical wobble

Unit tangent bundle of the Poincaré disc T 1(D).

Helical wobble

Unit tangent bundle of the Poincaré disc T 1(D).

Helical wobble

Unit tangent bundle of the Poincaré disc T 1(D).

Main open problem

Suppose that the Godbillon-Vey invariant of a codimension 1 foliation on a 3-manifold is 0. Does that imply that the foliation is cobordant to zero ? gv : Cobordism(Foliations on 3 manifolds) → R

Main open problem

Suppose that the Godbillon-Vey invariant of a codimension 1 foliation on a 3-manifold is 0. Does that imply that the foliation is cobordant to zero ? gv : Cobordism(Foliations on 3 manifolds) → R

Theorem (Thurston) 1972 If M is a circle bundle over a compact surface, every codimension 1 foliation on M with no compact leaf can be isotoped to a foliation transversal to the fibers, therefore associated to a group of diffeomorphisms of the circle.

Haefliger Γ-structures

Codimension q Haefliger Γ-structure on a manifold X : An open covering Ui of X. Continuous maps fi : Ui → Rq, A cocycle θi,j of C ∞ diffeomorphisms of open sets of Rq such that θj,k ◦ θi,j = θi,k where it is defined and fj = θi,j ◦ fi. André Haefliger (1970) There exists a classifying space BΓ∞

q .

Every codimension q Γ-structure is the pull-back of a universal structure by some map f : X → BΓ∞

q .

Haefliger Γ-structures

Codimension q Haefliger Γ-structure on a manifold X : An open covering Ui of X. Continuous maps fi : Ui → Rq, A cocycle θi,j of C ∞ diffeomorphisms of open sets of Rq such that θj,k ◦ θi,j = θi,k where it is defined and fj = θi,j ◦ fi. André Haefliger (1970) There exists a classifying space BΓ∞

q .

Every codimension q Γ-structure is the pull-back of a universal structure by some map f : X → BΓ∞

q .

Existence of codimension q foliations

Theorem (Thurston) 1973 : A codimension q ≥ 2 Γ-structure on a compact manifold M is homotopic to a foliation if and only if its (abstract) normal bundle embeds in the tangent bundle of M.

jiggling

Existence of codimension 1 foliations

Theorem (Thurston) 1973 : Every C ∞ hyperplane field is homotopic to a foliation.

Homotopy type of the classifying space : Mather et Thurston

Theorem 1973 : There exists a “natural” continuous map B Diff r

c (Rq) → Ωq(BΓr q)

inducing an isomorphism in integral homology. Corollaries : Every plane field, in any dimension, is homotopic to a C 0 foliation. Cobordism(Foliations on 3 manifolds) ≃ H3(BΓ∞

1 , Z) ≃

H2(Diff∞

c (R), Z).

Homotopy type of the classifying space : Mather et Thurston

Theorem 1973 : There exists a “natural” continuous map B Diff r

c (Rq) → Ωq(BΓr q)

inducing an isomorphism in integral homology. Corollaries : Every plane field, in any dimension, is homotopic to a C 0 foliation. Cobordism(Foliations on 3 manifolds) ≃ H3(BΓ∞

1 , Z) ≃

H2(Diff∞

c (R), Z).

Warwick, Summer 76

On proofs and progress in mathematics, Thurston 1994

“I believe that two ecological effects were much more important in putting a damper on the subject than any exhaustion of intellectual resources that occurred. First, the results I proved [...] were documented in a conventional, formidable mathematician’s style. They depended heavily on readers who shared certain background and certain insights. [...] The papers I wrote did not (and could not) spend much time explaining the background culture. They documented top-level reasoning and conclusions that I often had achieved after much reflection and effort.”

On proofs and progress in mathematics, Thurston 1994

“Second is the issue of what is in it for other people in the

• subfield. When I started working on foliations, I had the

conception that what people wanted was to know the

• answers. I thought that what they sought was a collection
• f powerful proven theorems that might be applied to

answer further mathematical questions. But that’s only

• ne part of the story. More than the knowledge, people

want personal understanding. And in our credit-driven system, they also want and need theorem-credits.”

Back to Godbillon-Vey invariant.

What is the “qualitative meaning” of a non zero Godbillon-Vey number ? Suppose two codimension one foliations of class C ∞ on a 3 manifold are topologically equivalent. Do they have the same Godbillon-Vey number ?

Godbillon-Vey : some kind of self linking number of a foliation ?

Dennis Sullivan : Let F be a codimension 1 foliation on M3. Choose a flow φt transverse to the foliation. Think of F as a 2-current : approximate by a large number of large balls in leaves. Compute link(F, φt(F) =

• M dω ∧ (φt)⋆ω

gv(F) = d2 dt2 link(F, (φt)⋆(F))|t=0

Godbillon-Vey : some kind of self linking number of a foliation ?

Dennis Sullivan : Let F be a codimension 1 foliation on M3. Choose a flow φt transverse to the foliation. Think of F as a 2-current : approximate by a large number of large balls in leaves. Compute link(F, φt(F) =

• M dω ∧ (φt)⋆ω

gv(F) = d2 dt2 link(F, (φt)⋆(F))|t=0

Godbillon-Vey and resilient leaves

Theorem (Duminy) 1982 : If gv(F) = 0, there is a resilient leaf.

Godbillon-Vey and projective group

Let R : π1(Σ) → Diff∞

+ (S1).

Obstruction to be projective in schwarz(R) ∈ H1(Diff∞

+ (S1), {u(x)dx2})

If Rt depends on a parameter, dRt dt ∈ H1(Diff∞

+ (S1), {v(x) ∂

∂x }) Pairing H1(π1(Σ)), {u(x)dx2}) ⊗ H1(π1(Σ), {v(x) ∂ ∂x }) → H2(π1(Σ), {w(x)dx}) → R

Godbillon-Vey and projective group

Let R : π1(Σ) → Diff∞

+ (S1).

Obstruction to be projective in schwarz(R) ∈ H1(Diff∞

+ (S1), {u(x)dx2})

If Rt depends on a parameter, dRt dt ∈ H1(Diff∞

+ (S1), {v(x) ∂

∂x }) Pairing H1(π1(Σ)), {u(x)dx2}) ⊗ H1(π1(Σ), {v(x) ∂ ∂x }) → H2(π1(Σ), {w(x)dx}) → R

Godbillon-Vey and projective group

Let R : π1(Σ) → Diff∞

+ (S1).

Obstruction to be projective in schwarz(R) ∈ H1(Diff∞

+ (S1), {u(x)dx2})

If Rt depends on a parameter, dRt dt ∈ H1(Diff∞

+ (S1), {v(x) ∂

∂x }) Pairing H1(π1(Σ)), {u(x)dx2}) ⊗ H1(π1(Σ), {v(x) ∂ ∂x }) → H2(π1(Σ), {w(x)dx}) → R

Godbillon-Vey and projective group

Let R : π1(Σ) → Diff∞

+ (S1).

Obstruction to be projective in schwarz(R) ∈ H1(Diff∞

+ (S1), {u(x)dx2})

If Rt depends on a parameter, dRt dt ∈ H1(Diff∞

+ (S1), {v(x) ∂

∂x }) Pairing H1(π1(Σ)), {u(x)dx2}) ⊗ H1(π1(Σ), {v(x) ∂ ∂x }) → H2(π1(Σ), {w(x)dx}) → R

Godbillon-Vey and projective group

Theorem (Maszczyk) 1999 d gv(Rt) dt = schwarz(Rt).dRt dt

Gelfand and Fuchs simple model : "piecewise projective foliations"

Start with a simplicial complex. Foliate each simplex by a pencil of hyperplanes containing a codimension 2 subspace, disjoint from the simplex. All these foliations should be coherent on boundaries of simplices.

Gelfand and Fuchs simple model : "piecewise projective foliations"

Gelfand and Fuchs simple model : "piecewise projective foliations"

Gelfand and Fuchs simple model : "piecewise projective foliations"

Theorem (Gelfand and Fuchs) There is a classifying space BPL. There is a non trivial “Godbillon-Vey invariant” H3(BPL, R).

A combinatorical cocyle for "piecewise projective foliations"

Rogers L function for 0 < x < 1. L(x) = −1 2 x ln(1 − t) t + t 1 − t

• dt − π2

6 Theorem The Gelfand-Fuchs-Godbillon-Vey invariant on piecewise projective foliations is represented by L evaluated on the cross ratio of four hyperplanes. H3(BPL, Z) → R is injective !

A combinatorical cocyle for "piecewise projective foliations"

Rogers L function for 0 < x < 1. L(x) = −1 2 x ln(1 − t) t + t 1 − t

• dt − π2

6 Theorem The Gelfand-Fuchs-Godbillon-Vey invariant on piecewise projective foliations is represented by L evaluated on the cross ratio of four hyperplanes. H3(BPL, Z) → R is injective !

Thurston cocycle on the group of diffeomorphisms of the circle

Thurston(f , g, h) =

• S1
• 1

ln Df d ln Df 1 ln Dg d ln Dg 1 ln Dh d ln Dh

• dt

is a homogeneous 2-cocycle on Diff which represents the Godbillon-Vey class.

What is the “natural” domain of definition of Godbillon-Vey ?

• S1 x(t) dy(t) is the area of a curve (x(t), y(t)) in the plane.

C 2 foliations. f of class C 1 such that ln Df has bounded variation (Duminy and Sergiescu). f is of class C 1+α with α > 1/2 (Katok-Hurder) BΓ1+α

1

is contractible if α < 1/2 (Tsuboi).

What is the “natural” domain of definition of Godbillon-Vey ?

• S1 x(t) dy(t) is the area of a curve (x(t), y(t)) in the plane.

C 2 foliations. f of class C 1 such that ln Df has bounded variation (Duminy and Sergiescu). f is of class C 1+α with α > 1/2 (Katok-Hurder) BΓ1+α

1

is contractible if α < 1/2 (Tsuboi).

What is the “natural” domain of definition of Godbillon-Vey ?

• S1 x(t) dy(t) is the area of a curve (x(t), y(t)) in the plane.

C 2 foliations. f of class C 1 such that ln Df has bounded variation (Duminy and Sergiescu). f is of class C 1+α with α > 1/2 (Katok-Hurder) BΓ1+α

1

is contractible if α < 1/2 (Tsuboi).

What is the “natural” domain of definition of Godbillon-Vey ?

• S1 x(t) dy(t) is the area of a curve (x(t), y(t)) in the plane.

C 2 foliations. f of class C 1 such that ln Df has bounded variation (Duminy and Sergiescu). f is of class C 1+α with α > 1/2 (Katok-Hurder) BΓ1+α

1

is contractible if α < 1/2 (Tsuboi).

What is the “natural” domain of definition of Godbillon-Vey ?

• S1 x(t) dy(t) is the area of a curve (x(t), y(t)) in the plane.

C 2 foliations. f of class C 1 such that ln Df has bounded variation (Duminy and Sergiescu). f is of class C 1+α with α > 1/2 (Katok-Hurder) BΓ1+α

1

is contractible if α < 1/2 (Tsuboi).

Malliavin-Shavgulidze “Haar measure”.

How to choose a diffeomorphism of the circle “at random” ? Choose it so that the log of its derivative is a random function on the circle. Choose a random path t ∈ [0, 1] → u(t) ∈ R with u(0) = 0. Transform it into a random bridge t ∈ [0, 1] → b(t) = u(t) − tb(1), so that b(0) = b(1) = 0. Define a random diffeomorphism of the circle R/Z by f (t) = f (0) + exp( t

0 b(t) dt)

exp( 1

0 b(t) dt)

where f (0) is random with respect to the Lebesgue measure. This defines the Malliavin-Shavgulidze probability measure on Diff+

1 (S1).

Almost surely, the derivative of a circle diffeomomorphism is Holder 1/2.

Malliavin-Shavgulidze “Haar measure”.

How to choose a diffeomorphism of the circle “at random” ? Choose it so that the log of its derivative is a random function on the circle. Choose a random path t ∈ [0, 1] → u(t) ∈ R with u(0) = 0. Transform it into a random bridge t ∈ [0, 1] → b(t) = u(t) − tb(1), so that b(0) = b(1) = 0. Define a random diffeomorphism of the circle R/Z by f (t) = f (0) + exp( t

0 b(t) dt)

exp( 1

0 b(t) dt)

where f (0) is random with respect to the Lebesgue measure. This defines the Malliavin-Shavgulidze probability measure on Diff+

1 (S1).

Almost surely, the derivative of a circle diffeomomorphism is Holder 1/2.

Malliavin-Shavgulidze “Haar measure”.

How to choose a diffeomorphism of the circle “at random” ? Choose it so that the log of its derivative is a random function on the circle. Choose a random path t ∈ [0, 1] → u(t) ∈ R with u(0) = 0. Transform it into a random bridge t ∈ [0, 1] → b(t) = u(t) − tb(1), so that b(0) = b(1) = 0. Define a random diffeomorphism of the circle R/Z by f (t) = f (0) + exp( t

0 b(t) dt)

exp( 1

0 b(t) dt)

where f (0) is random with respect to the Lebesgue measure. This defines the Malliavin-Shavgulidze probability measure on Diff+

1 (S1).

Almost surely, the derivative of a circle diffeomomorphism is Holder 1/2.

Malliavin-Shavgulidze “Haar measure”.

How to choose a diffeomorphism of the circle “at random” ? Choose it so that the log of its derivative is a random function on the circle. Choose a random path t ∈ [0, 1] → u(t) ∈ R with u(0) = 0. Transform it into a random bridge t ∈ [0, 1] → b(t) = u(t) − tb(1), so that b(0) = b(1) = 0. Define a random diffeomorphism of the circle R/Z by f (t) = f (0) + exp( t

0 b(t) dt)

exp( 1

0 b(t) dt)

where f (0) is random with respect to the Lebesgue measure. This defines the Malliavin-Shavgulidze probability measure on Diff+

1 (S1).

Almost surely, the derivative of a circle diffeomomorphism is Holder 1/2.

Malliavin-Shavgulidze “Haar measure”.

How to choose a diffeomorphism of the circle “at random” ? Choose it so that the log of its derivative is a random function on the circle. Choose a random path t ∈ [0, 1] → u(t) ∈ R with u(0) = 0. Transform it into a random bridge t ∈ [0, 1] → b(t) = u(t) − tb(1), so that b(0) = b(1) = 0. Define a random diffeomorphism of the circle R/Z by f (t) = f (0) + exp( t

0 b(t) dt)

exp( 1

0 b(t) dt)

where f (0) is random with respect to the Lebesgue measure. This defines the Malliavin-Shavgulidze probability measure on Diff+

1 (S1).

Almost surely, the derivative of a circle diffeomomorphism is Holder 1/2.

Theorem (Malliavin / Shavgulidze) :

This probability measure is quasi-invariant under left translations by C 3-difffeomorphism. d(Lφ)⋆µ dµ (f ) = exp

• S1 Sφ(f (t))(f ′(t))2dt

Michele Triestino

Michele Triestino

Using stochastic integration, the Thurston cocycle can be defined almost everywhere in Diff1

+(S1) and defines a measurable

Godbillon-Vey cohomology class.

• B1(t) dB2(t)

Problem : Compute the “measurable Gelfand-Fuchs cohomology” Diff1

+(S1) with respect to Malliavin-Shavgulidze measure.

Michele Triestino

Using stochastic integration, the Thurston cocycle can be defined almost everywhere in Diff1

+(S1) and defines a measurable

Godbillon-Vey cohomology class.

• B1(t) dB2(t)

Problem : Compute the “measurable Gelfand-Fuchs cohomology” Diff1

+(S1) with respect to Malliavin-Shavgulidze measure.

A lost theorem of Thurston ?

Theorem ? ? ? ? ? ? ? The cube of the Euler class eu ∈ H2(Diff+

analytic(S1), Z)) vanishes.

A lost theorem of Thurston ?

Theorem ? ? ? ? ? ? ? The cube of the Euler class eu ∈ H2(Diff+

analytic(S1), Z)) vanishes.

“I forgot !”